Question:
"What condition must be met by the two vectors of a basis such that any given vector can be expressed as a linear combination of these two vectors?"
Answer (from answer booklet):
"They must be linearly independent (or non-collinear)."
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Before checking the answer, I answered the exact opposite. My argument is that, when you combine two vectors in order to form another with a linear combination, those two vectors had to be collinear in order for them to produce another vector in the same linear path (=same slope).
How am I wrong?
Yes, "in order for them to produce another vector in the same linear path" but the question was for them to produce any given vector. For example, the vectors (1, 1) and (2, 2) are collinear and any linear combination of them: a(1,1)+ b(2,2)= (a+ 2b, a+ 2b) is of the form (x,x). The vector (1, 3) cannot be written as a linear combination of them.
But I would have another objection to that answer. Two independent vectors will form a basis in 2 dimensions, but that is not said by the problem. In fact the problem say "two vectors of a basis" which seems to imply that there may be more vectors in that same basis. But certainly any two vectors in a basis must be linearly independent. In order that any vector could be written as a linear combination of them, the vectors must span the space- they must be the only vectors in that basis.